Ultramicroscopy 6 (1981) 29-40North-Holland Publishing Company

LATTICE SPACINGS FROM LATTICE FRINGES

P.G. SELF, H.K.D.H. BHADESlliA and W.M. STOBBSDepartment of Metallurgy and Materials Science, University of Cambridge, Pembroke Street, Cambridge CB2 3QZ, England

Received 21 October 1980

The different techniques available for the accurate measurement of lattice fringe spacings are discussed and their rela-tive merit under different circumstances assessed. The relationship between the lattice fringe spacing and the lattice param-eter is further examined for both uniform and chemically inhomogeneous material. The techniques described were alsoapplied to an investigation of carbon segregation in residual austenite after the upper bainitic transformation in two steels.

1. Introduction and reviewed by Sinclair [6]. It involves the measure-ment of lattice fringe spacings and their correlationwith lattice spacing variation due to such effects assegregation. Our aim here is to delineate the accuracyof possible measurement methods and to indicate thefactors which determine the accuracy of the techni-que when applied to different physical systems. Atthis stage it should be pointed out that the techniqueis complimented by the methods of convergent-beamelectron diffraction [7,8] which are of at least as higha potential accuracy in situations where a'convergent-beam probe can be used on the specimen area ofinterest.

The relative accuracy of the various differentmethods for measuring spacings in a micrograph aredescribed briefly in section 2 while the relativeimportance of the various ways in which such ameasurement need not reflect the true lattice spacing.is discussed in section 3. An example of the applica-tion of the method to the examination of the coarsesegregation of carbon from upper bainitic ferrite toaustenite in two steels, given in section 4, is followedby a summary of our conclusions in section 5.

2. Fringe spacing measurement techniques

Modem transmission electron microscopes may beused routinely to obtain images with point-to-pointresolution of well under 0.3 nm and axial latticefringe images showing spacings of less than 0.2 nm.However, in the majority of high resolution work thefine detail in the micrographs (such as absolute inten-sities and absolute fringe spacings) has not been fullyexploited. Rigorous analyses have sometimes beencompleted successfully (e.g. O'Keefe and Buseck [1])but in most cases full matching of fine detail withimage calculations has proved impossible (e.g. Bursill,Barry and Hudson [2] and Anstis et al. [3]). Theseexamples show that, unless the structural features ofinterest are relatively coarse compared with theattainable resolution, considerable care must be takenin the interpretation of images. Generally, even afterfull image simulation and quantitative matching ofthe computed and experimental micrographs, thestructural featu~es remain ambiguous. This is becauseof uncertainty in instrumental parameters (Cs' Ccetc.), the speciinen thickness and also the verynature of electron scattering. Saxton [4] has reviewedsome of the techniques which can be applied for thecorrection of artifacts in both linear and non-linearimages.

A now quite generally applied method whichpromises material information with relative ease isthat employed recently by Sinclair and Thomas [5]

There are several methods available for measuringlattice fringe spacings and it is not easy to tell whichis the most accurate for a given system. The advan-

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30 P. G. Self et al. / Lattice spacings from lattice fringes

tages and disadvantages of three techniques are sum-marised in this section. A feature common to anymethod is the need for a magnification calibrationeven if this is only indirect: an internal standard ofknown lattice spacing being available in the specimenof interest. If this is not the case the microscope mag-nification must be accurately determined as a func-tion of specimen height using the objective lenscurrent as a scale. In fact very small height changescan cause significant magnification changes. Forexample, using the short focal length JEaL Cs = 0.7mm lens, a height change of only 10 pm (as producedby a specimen tilt of one degree for a specimen 0.5mm from the eucentric point ofa specimen holder)gives as much as a 1 % change in magnification.

2.1. Moire fringes

A simple and quick way of determining fringespacings is to compare the fringes with a ruled gratingby forming moire fringes. The accuracy of the techni-que is generally insufficient for most applications,being limited both by the small number and breadthof the moire fringes which can be used in the mea-surement. The usefulness of-the method lies in the

ease with which suitable areas for examination bymore accurate methods can be located and micro-graphs giving irregular moire fringes discarded.

If the spacing of lattice fringes is to be measuredaccurately, it is necessary to have large regions offringes with uniform spacing. Any deviation fromconstant spacing may cause a significant error in theaverage fringe spacing obtained. Even micrographs ofareas containing no crystallographic defects can oftenhave changes in fringe orientation or spacing. This isin general associated with small changes in focus,specimen thickness and orientation across the speci-men. Fig. I shows (Ill) fringes in aluminium.Although at first sight the fringes appear uniform themoire fringes formed are curved. The largest regionswhich did not exhibit curved fringes were typically300 fringe spacings in extent. Hence, in this particularcase, although there were a thousand or so fringes onthe micrograph, the accuracy is limited to that attain-able using only 300 fringes.

A good way of applying the moire technique is toproject the micrograph onto a ruled grating. The typi-cally low contrast levels of lattice fringe micrographscan be balanced to give optimum moire fringe con-trast by using a grating with dark to light line width

Fig. 1. An area of (Ill) fringes from aluminium (a) which appear straight yet which when compared with a ruled grating theresultant moire fringes show a marked curvature (b).

P..G.. Self et 01.. / Lattice spacings from lattice fringes 31

ratio of about 1 to 3. Even so the moire fringes aredifficult to record.

2.2. Microdensitometry

The principle of micro densitometric techniques isto count the number of fringes in a measured dis-tance. This can be done in several ways using, forexample, a travelling table microdensitometer, atravelling table microscope or computer processing. Itis obviously essential that there are no defects alongthe measured line, and so a check of the micrographby the moire method before proceeding with thisoften time consuming process is particularly useful.

Systematic error can be introduced into this tech-nique by measuring along a line not perpendicular tothe fringe direction. This can be eliminated by using arotating table and measuring the fringe spacing for setmisorientations. The true fringe spacing is then deter-mined by finding the interpolated minimum value.Another source of systematic error can arise if thefringe shape is different at the two ends of themeasured line, but this is usually detectable. Withreasonable care the technique can be used to give thespacing of a set of fringes to an accuracy of 0.01 % orbetter.

Another attractive technique under this generalheading is to digitize the intensities on the micro-graph and process them in a computer to give thefringe spacing using programs such as the SEMPERsystem [9]. The method is however time consumingand possible marginal improvements in accuracy aregenerally not warranted, given sources of error in theway the fringe spacing relates to that of the lattice.

In some applications it is necessary to measure thevariation in spacing from fringe to fringe, as in thestudy of the metallurgically interesting phenomena ofspinodal decomposition [5,10]. In this case the onlysuitable technique is microdensitometric. The accur-acy is however then relatively low, mainly because ofthe difficulty in locating equivalent measuring pointson each fringe - the contrast being locally variable.

2.3. Optical transfonnation

review see Mulvey [11]). The technique's principalmerit lies in the way an average for a large region isobtained directly giving potentially a higher accuracythan the other methods described. Even when exa-mining small regions, provided they contain uniformfringes, the centres of the optically diffracted spotscan be located with precision to give a value for thelattice fringe spacing to very high accuracy. Suchmeasurements require, of course, that the opticalbench conditions be optimised for spot measurement,and abberations in the optical system be minimized.

Experimentally the technique is most accuratewhen high-order spots can be used in the measure-ment. Reprinting the original micrograph onto highcontrast fIlm is sometimes advantageous (particularlyif the original micrograph has low contrast) becausethe resultant squaring of the fringe contrast gives riseto higher-order spots. Generally, however, onlysecond-order spots are produced in this way, mainlybecause of the marked variation in the width of thedark and light regions of a fringe along its length.

One advantage of the optical technique is that it ispossible to have a defect in the area of the micro-graph used to form the diffraction pattern and stillget better accuracy than when examining a smallerdefect-free region. This is provided that the defect isnot associated with any fringe rotation except in alocalised region of low contrast. The effect of defectson the optical diffractogram will vary from system tosystem, and so when measuring over a region knownto contain a defect it is best to repeat, the procedurefor a defect-free region (necessarily of smaller size) toensure that improved accuracy has in fact beenobtained.

The above advantage of the technique also reflectsits main disadvantage. This is that, althou~ theaverage spacing of the fringes over the whole areaexamined may be found very accurately, the uncer-tainty in this value is difficult to determine. The onlyway of ascertaining the spread of spacing in themicrograph is by careful analysis of the diffractedspot intensities and shapes. In practice it is usuallyfound that an examination of a through focal seriesbest delineates the overall uncertainty in themeasured fringe spacing (see section 3). Examples oftypical diffractograms for systems varying from idealto real structures are shown in fig. 2.

A standard method used in the measurement ofspatial frequencies in a micrograph is to obtain itsFourier transform using an optical bench (for a

32 P. G. Self et al. / Lattice spacings from lattice fringes

Fig. 2. Optical transfonns for several systems: (a) the undiffracted beam, (b) a rule grating, (c) a micrograph with low contrastfringes, (d) contrast enhanced fringes and (e) a high defect density material.

3. The relation between fringe spacing and latticespacing

spacings on a single micrograph to a measuring accur-acy of 0.01 % it is necessary to quantify how well themeasured fringe spacing reflects the true latticespacing. We have already noted some of the reasonsGiven that it is possible to measure lattice fringe

P.G. Self et al. / Lattice spacings from lattice fringes 33

1.005

Q)

0u

U")

.6'-

«C7' 1.000cu0c..

U")

Q)C7'0'-Q)>

-« 0.9' 5

b

Fig. 3. The fringes from a high Otromium steel (a) and their spacing as measured on a travelling table microscope ,(b). As morefringes are counted the measuring accuracy increases but, as shown, the uncertainty in the true fringe spacing remains relativelyhigh.

why the fringe spacing can vary locally from ~e truelattice spacing (see section 2.1). A single micrographof even a perfect crystal will thus show variable spac-ings from point to point. It is this that will1imit theaccuracy of the fringe spacing measurement ratherthan the measuring accuracy of the technique used.This is clearly demonstrated in fig. 3 where spacingmeasurements are shown for a high purity Fe-Cr-Nialloy. The plot shows that as the number of fringesused in the measurement is increased, and hence themeasuring accuracy (indicated by the error bars)decreases, the spread in measured fringe spacingremains well above that to be expected from the

measuring accuracy.

By considering the effects giving rise to local fringespacing variation it becomes clear that regionsshowing variability will change as the objective lenstransfer function is altered. Table 1 shows the resultsof fringe spacing measurements from a through focalseries of the specimen area shown in fig. 3. It can beseen that the variation in fringe spacing is much largerthan that expected from the measuring error of anysingle micrograph which was 0.005%. Hence in orderto obtain a more realistic uncertainty in a latticefringe measurement it is necessary to use the resultsobtained from a through focal series of the sameregion of specimen.

One of the results in table I is anomalously low.

34 P.G. Self et al. / LI1ttice spacings from lattice fringes

resultant lattice fringe spacing can be very differentfrom the lattice periodicity and change with thecrystal tilt and defocus. It may even depend onspatial coherence. As an example, fig. 4 shows theamplitude profile of a diffraction spot broadened as itwould be for a specimen area 20 nm in extent(as (sin x)/x). The profiles are computed assuming aweak phase object for a 0.7 mm Cs lens and a latticeof 0.4 nm if viewed axially orO.2 nm if viewed withsymmetrically tilted illumination. It may be seen thatnear to Scherzer defocus (- -70 nm) the amplitudeis transferred relatively well but is severely distortedat some defoci, especially when the transfer functionfor the relevant angle is small. Note, for example, thatwhen ~f= -190 nmthe fringe spacing observedwould be at least 1 % smaller than the lattice periodi-city. It is significant that the observed change in spac-ing for weak transfer at moderate defocus (table 1) isof similar magnitude and sign to that predicted herethough, in general, whether the spacing increases ordecreases is a function of both lattice periodicity and

1.70861.71041.70941.70971.7078*

1.7095 t 0.0005 (0.03%)

This value was obtained from a micrograph with veryweak fringe contrast, so that the focus value musthave been such that the objective lens transfer func-tion was close to zero at the Bragg angle. Thismeasurement demonstrates a very important effect infringe measurement: if the spots in the electrondiffraction pattern from the crystal are broad, the

A

1"\Or-~ \/"' """/~

'-/

50 -70

I/""""o~ "\~ -""'J -- ~I'

J\ 11'0 -130~

('

o~ f"/"'-

\)-190-176

-I +

4.6

+'

5.0+ +

5.4 4.6+-

5.0

+ +5.4 4.6

+

5.0

+5.4

( ~l )nm

Fig. 4. The effect of lens transfer on a shape-broadened diffraction spot. The resultant spot shape is calculated at several defocifora Cs = 0.7 mm lens. The system is for a 20 nm area of 0.2 nm fringes with tilted illumination.

P. G. Self et al. / Lattice spacings from lattice fringes 35

and Thomas [13]. Spence et al. modelled the latticeperiodicity as a sine function and showed that thelocal fringe spacing should vary with focus and that itcan be drastically different (10-15%) from the modelspacing, even at optimum defocus. However, theywere unable to explain the large spread in spacingobserved by Gronsky and Thomas.

While it thus appears that the effect of the transferfunction alone is sufficient to explain Gronsky andThomas' result it is important to realise that Spenceet al.'s arguments apply equally well to a system withuniform lattice spacing but sinusoidally varyingprojected lattice potential. Spinodal decompositionis of course associated with segregation and conse.quent changes in a local structure factor. These neces-sarily give rise to local variations in the amplitude andphase of the scattered waves. While the variableamplitude results in changes in contrast, phase varia-tions necessarily cause a change in local fringe spac-ing. The effect can either increase or decrease thespacings observed on a local scale, though an increasewould be generally expected in that higher atomicnumber elements normally segregate with associatedincreases in both lattice parameters and scatteringfactor.

Further problems include the fact that it isunlikely that spinodal decomposition is associatedwith a simple sinusoidal modulation of lattice spacing[14,15] . An effect of probably even greater import-ance is that the segregation in a thin foil will result insurface relaxations and further lattice distortions,exacerbated by the three-dimensional nature ofsegregation. This effect has been discussed by Cookand Howie [16] in another connection, and Howie[17] has recently pointed out its significance here.

Although the nature of all these effects is differentthe change required to allow for them in the two-beam analysis used by Spence et al. [12] is the same.Their equation for the two-beam intensity is

I(x) = l<I>of + 21<1>0 II <l>u' (x)1 COS[21TU~X + <t>(X)]

(1)

where <Po and <Pu' are the amplitude of the zerothorder and diffracted beam respectively. The locallattice spacing is liu~ and cf>(x) is the phase differencebetween the zeroth order and diffracted beam(including that introduced by the lens system). The

defocus. These calculations also make it clear that theappropriate defocus for accurate transfer becomesmore critical for smaller fringe spacing, especially ifthis is beyond the first zero of the transfer function.

The diffraction spots can be broad for a number ofreasons. The most significant reason for broadening,and one which is always present irrespective of thesystem, is due to the specimen being of finite thick-ness. In conditions where the crystal is tilted awayfrom the Bragg angle or the surface normal is at ahigh angle to the lattice planes, the fringe spacing willbe markedly different from the true lattice periodi-city. Table 2 shows the results of the fringe measure.ment (using a through focal series) from a spatiallylimited area of uniform fringes. The system was amartensitic memory alloy (Cu-Zn-Al) of monoclinicstructure and the fringes used were of approximately0.6 nm spacing G (001)) and were observed in aregion limited by two (001) plane stacking faultsabout 20 nm apart. The spread in the measured fringespacing is some four times higher than the measuringerror and again as would be expected with referenceto the above calculation.

Just as when. local irregular variations make theobserved spacing sensitive to defocus we again con-clude that to obtain a reasonable value of a latticespacing it is necessary to measure a.through focalseries of the same region.

Situations in which observed fringe spacings haveto be used even more carefully arise when theperiodicity is locally variable. This is the case forspino dally decomposed systems. Spence et al. [12Jhave considered the effect of the lens transferfunction for the Au-Ni system examined by Gronsky

36 P.G. Self et al. / Lattice spacings from lattice fringes

4. Application of the method to carbon segregation intwo steels

spatial coordinate perpendicular to the fringes is x.All that is required to take into account the effectsdiscussed above is to make the phase and amplitudesof the beams a function of x. The two-beam solutionfor the intensity becomes:

I(x) = l<I>o(x)f + 21<I>o(x)11 <I>u' (x) I

X COS[21TU~X + ct>(x)] . (2)

In this section we examine the accuracy of thetechnique when applied to a real problem: the segre.gation of carbon from ferrite to austenite in twosteels. The compositions of the two steels examinedwere (in wt%):

Alloy 1: 3% Mn, 2.02% Si, 0.43% C and remainderFe,Alloy 2: 4.08% Ni, 2.05% Si, 0.39% C and remainderFe.

This is more difficult to analyse than the originalexpression and a full solution would require bothcareful choice of model and comprehensive machineanalysis. However, the important effects on the fringeprofIles can be deduced intuitively.

Consider, for example, two adjacent areas withuniform but different lattice spacing. These areas willgive rise to different scattered amplitudes and phases.Both will show uniform fringe spacings but thefringes will be shifted relative to the local latticeplane positions differently in each area. The interfacebetween the two regions will thus show a largediscrepancy in the local fringe spacings. Examinationof the fringe profIles reported by Gronsky andThomas [13] do indeed suggest that the majoranomaly is the grouping of a few fringes of extremelysmall spacing as the above qualitative argument wouldindicate remembering that the discrepancy will befurther compounded by the effects of defocus.

On the same approach it is clear that as the objec-tive lens focus is changed the regions where thefringes are more uniformly spaced will show littlechange in spacing but rather a bulk shift of thefringes. At the modelled block interface the fringespacing could vary quite markedly. This places doubton the practice of averaging over a few fringe spacingsto increase accuracy, but on the other hand suggestsan improvement to the method for determining locallattice spacings. If the fringe profIles are plotted foreach micrograph of a through focal series of imagesthe local regions where the fringe spacing stays rela-tively constant should more accurately reflect thelocal lattice spacings. Gronsky and Thomas' fringeprofIle thus suggest the importance of including non-sinusoidal terms in the modelled expressions for thespacings. Appropriate models could probably thus befound by comparison of dark-field images as a func-tion of deviation parameter.

The fonner alloy was austenitised at 1100°C for10 min, isothermally transformed at 350°C for 1000min and then water quenched. The latter material wassimilarly austenitised but isothermally transformed at340°C for 60 min before water quenching. These heattreatments place the alloys in the upper bainite trans-fonnation temperature range and the time periods ofthe isothermal treatments ensure reaction terminationin both cases [18].

The bainite transfonnation in steels is a displacivereaction [19,20] in which the ferrite (a) is initiallysupersaturated with respect to carbon. In the absenceof carbide precipitation (prevented in the presentsteels by the use of silicon), the excess carbon in thebainitic ferrite is subsequently rapidly partitionedinto the residual austenite ("1). The further transfor-mation of this residual austenite to bainitic ferriteceases when the carbon content of the fonner reachesa level Xoy(T 0), such that displacive transfonnationbecomes thennodynamically impossible [20].Although such austenite can no longer transfonn tobainite, it can'continue to accumulate carbon fromsuitable sources up to a limit given by the no-substitu-tional partitioning Ae; curve [18-20], i.e. xoy(Ae;).Such circumstances arise naturally during bainitesheaf formation [20] when a region of austenite,which has been affected by the dumping of carbonfrom an extant bainite plate, becomes isolated by theformation of new supersaturated bainite plates in itsclose proximity. The subsequent partitioning ofcarbon from these initially supersaturated boundingplates can raise the carbon content of the entrappedaustenite film to any level within the range Xoy(T 0) ~Xoy ~ xoy(Ae;). The lower limit arises from the fact

37P. G. Self et al. / Lattice spacings from lattice fringes

Table 3Experimental results for the carbon content (X-y) of retained austenite in two steels (see text), the second column indicates thenumber of micrographs in the through focal series used in the measurement; note the marked variation in carbon content fromregion to region in both alloys

-Region Number

in seriesd(lll)'Y/d(O!l)Q

':X'Y(wt%)

Micro. Optical Micro. Optical

Alloy 1

1.029 j; 0.0011.033 j; 0.0021.046 j; 0.002

1.023-1.047

1.028:!: 0.0011.034 :!: 0.001

1.31 :I: 0.08 (6%)1.62:1: 0.16 (10%)2.66 :I: 0.16 (6%)

0.85-2.80

645

1.23 :t 0.08 (7%)1.70 :t 0.08 (5%)

,023-1.047 0.85-2.80

3453

1.030:1: 0.0031.040 :!: 0.005

1.044:!: 0.0011.076 :!: 0.002

1.028-1.053

1.39 f 0.24 (17%)2.18 f 0.40 (18%)2.50 f 0.08 (3%)5.06 f 0.16 (3%)

1.25-3.20

23

Predicted

Alloy 2

1234

Predicted 1.028-1.053 1.25-3.20

the alloys should have a constant lattice parameter of0.28664 nm [21]. Although two distinct experimen-tal relationships have been described for the latticeparameter of austenite as a function of carbon con-tent [22,23], it seems that the approach of Roberts[22] is more appropriate in the current application[18]. His expression for the austenite lattice param-eter, a-y is:

a-y = 0.3555 + 0.0044 x-y (nm) , (3)

where X-y is the wt% of carbon in austenite.Images from each material are shown in fig. 5

and each exhibits both (OII)a and (111)-y latticefringes. Qualitatively the defect density is high inboth materials and is apparently significantly higherin the austenite of alloy 2 which had a lower volumefraction of this phase. Correspondingly, the applica-tion of convergent-beam techniques was impractic-able and the application of lattice fringe measure-ment techniques is suitably difficult. In fact, thelattice fringe spacing of the retained austenite in alloy2 could not be measured by the optical technique.

The results from several through focal series for anumber of different regions of each alloy are given intable 3. Here, since the measurements made wererelative, only the ratios for the (OII)a and (111)-yspacings are shown. Absolute values may be deter-mined using the ferrite lattice parameter and

that any austenite with a carbon content less thanx-r<T 0) should ultimately transform to bainite. Itshould be noted that although substitutional alloyingelements do not partition during the bainite trans-formation, their influence on the free energies ofaustenite and ferrite is manifested in the x'Y(T 0) andx-r<Ae;) values. Thus any experimental measurementsshould reflect generally higher carbon levels in thecase of alloy 2 (see table 3). Accordingly the latticeparameters of the retained austenite in each alloy willbe different. On the other hand, since the fmalcarbon content of the bainitic ferrite is very small anddoes not vary significantly, the lattice parameter ofthe ferrite can be used as an internal relative standardfor both the sets of austenite lattice fringe spacingmeasurements.

The results described below thus test both thethermodynamic approach used and the predictions ofthe previous sections on the relationship betweenlattice fringe and lattice spacings. The two steels werespecifically chosen to have very different predictedaustenite carbon content limits in order better to testthe technique of estimating x'Y by lattice fringe mea-surements,

4.1. Results and discussion

For the reasons discussed above, the ferrite in both

38 P.G. Self et al. / Lattice spacings from lattice fringes

Fig. 5. Micrographs of the two steels used in section 4 showing fringes from the ferrite (c and d) and austenite (e and f). Alloy 1is on the left and alloy 2 on the right.

austenite carbon contents by using the relation givenabove. The best accuracy obtained for the ratio wasapproximately 0.1 % which is much worse than mightbe expected given the measurement accuracy for a

single micrograph but reflects the strong variation offringe spacing with focus and the small sizes of detect-free regions. This is particularly the case in alloy 2where it was very difficult to obtain accuracies better

P.G. Self et al. / Lattice spacings from lattice fringes 39

mining the lattice fringe to lattice spacing relationshipand suggest ways of experimentally investigating thiseffect.

Our experiments on the measurement of latticefringe spacings in steels containing retained austenitehad the dual aim of examining the evidence forcarbon segregation and testing the accuracy of themethods described on a system with a high disloca-tion content for which accuracies might be expectedto be low. Nevertheless we were able to demonstratethat carbon segregation affects the lattice parametersufficiently to test both thermodynamic models forthe upper bainitic transformation and to discover realnon-uniform carbon segregation.

Acknowledgements

The authors are grateful to Dr. A. Howie fordiscussion of some of the theory and to Dr. R.A.Ricks for supplying the micrographs used in section3. Financial support from the Science ResearchCouncil is also acknowledged.

than 0.3% because of the high defect density in theretained austenite.

The results show a large variation from region toregion. This cannot be explained in terms of any ofthe effects which can make the lattice fringe spacingdifferent from that of the lattice as described insection 3. There must thus be a real variation of the 'Ylattice spacing from region to region. The displacivenature of the bainitic transformation does not allowsubstitutional alloying element partitioning, so theregion variations cannot be linked with substitutionalelement segregation. While atom probe results(Bhadeshia and Waugh [24]) confirm that there is nosubstitutional element partitioning, the changes inlattice parameter that could be expected, even forgross Mn or Ni segregation, are anyway insignificant(see Pearson [21]).

It is thus clear that two predictions of the theorydiscussed above have been demonstrated:(1) the overall carbon content of the austenite inalloy 2 is generally higher than that in alloy 1;(2) in both cases the carbon is distributed veryinhomogeneously. Furthermore, all but one of theresults fall within the predicted x"f(To) to x~Ae;)range; the exception is not understood.

References

5. Conclusions

We have demonstrated that a lattice fringe spacingon a single micrograph of a typical uniform, defect-free material can be measured to an accuracy of0.01% or better using either microdensitometric oroptical diffraction techniques. We have also shownthe usefulness of moire methods for the detection ofsuitable regions for more accurate measurements. Ofmore interest we have delineated the extent to whichobjective defocus can cause variation in the observedlattice spacing to a different degree under variousobservational circumstances, and show how this canbe useful in obtaining a realistic value for the latticeparameters of a uniform material.

We have also considered how lattice fringe spacingscan be related to the variation of a lattice parameteras a function of regular segregation as occurs duringspinodal decomposition. We conclude that the varia-tion in structure factor from point to point is asimportant as the variability of lattice spacing in deter-

[1] M.A. O'Keefe and P. Buseck, Trans. ACA 15 (1979) 27.[2] L.A. Bursill, J.G. Barry and P.R.W. Hudson, Phil. Mag.

A37 (1978) 789.[3] G.R. Anstis, D.F. Lynch, A.F. Moodie and M.A.

O'Keefe, Acta Cryst. A29 (1973) 138.[4] W.O. Saxton, J. Microsc. Spectrosc. Electron., in press.[5] R. Sinclair and G. Thomas, Met. Trans. 9A (1978) 373.[6] R. Sinclair, in: Proc. 9th Intern. Congr. on Electron

Microscopy, Toronto, 1978, Vol. III, p. 140.[7] Roier, Acta Cryst. A25 (1969) 516.[8] J.W. Steeds, in: Introduction to Analytical Electron

Diffraction, Eds. J.J. Hren, J.I. Goldstein and D.C. Joy(Plenum, New York, 1979).

[9] W.O. Saxton, T.J. Pitt and M. Homer, Intramicroscopy4 (1979) 343.

[10] C.K. Wu, R. Sinclair and G. Thomas, Met. Trans. 9A(1978) 381.

[11] T. Mulvey, J. Microsc. 98 (1973) 232.[12] J .C.H. Spence, J .M. Cowley and R. Gronsky, Intra-

microscopy 4 (1979) 429.[13] R. Gronsky and G. Thomas, in: Proc. Modulated Struc-

tures Conf., Kailva Kona, Hawaii (Am. Inst. Phys., NewYork, 1979).

[14] J.W. Cahn and J.E. Hilliard, Acta Met. 19 (19710 151.[15] J.S. Langer, Ann. Phys. (NY) 65 (1971) 53.

P.G. Self et al. / Lattice spacings from lattice fringes40

[21] W.B. Pearson, A Handbook of Spacings and Structuresof Metals and Alloys, Vol. 2 (Perganlon, Oxford, 1967).

[22] C.S. Roberts, Trans. AIME 197 (1953) 203.{23] N. Ridley, H. Stuart and L. Zwell, Trans. AIME 245

(1969) 1834.[24] H.K.D.H. Bhadeshia and R. Waugh, unpublished.

[16] R.F. Cook and A. Howie, Phil. Mag. 20 (1969) 641.[17J A. Howie, private communication (1980).[18] H.K.D.H. Bhadeshia", Ph.D. Thesis, University ofCam-

bridge (1979).[19] H.K.D.H. Bhadeshia and D.V. Edmonds, Met. Trans.

lOA (1979) 895.[20] H.K.D.H. Bhadeshia and D.V. Edmonds, Acta Met. 28

(1980) 1265.